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Keywords:

  • Bolzano-Weierstraß principle;
  • cohesive principle;
  • MSC (2010) 03F60;
  • 03D80;
  • 03B30

Abstract

The aim of this paper is to determine the logical and computational strength of instances of the Bolzano-Weierstraß principle (BW) and a weak variant of it.

We show that BW is instance-wise equivalent to the weak König’s lemma for Σ01-trees (Σ01- WKL). This means that from every bounded sequence of reals one can compute an infinite Σ01-0/1-tree, such that each infinite branch of it yields an accumulation point and vice versa. Especially, this shows that the degrees d ≫ 0′ are exactly those containing an accumulation point for all bounded computable sequences.

Let BWweak be the principle stating that every bounded sequence of real numbers contains a Cauchy subsequence (a sequence converging but not necessarily fast). We show that BWweak is instance-wise equivalent to the (strong) cohesive principle (StCOH) and—using this—obtain a classification of the computational and logical strength of BWweak. Especially we show that BWweak does not solve the halting problem and does not lead to more than primitive recursive growth. Therefore it is strictly weaker than BW. We also discuss possible uses of BWweak. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim